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AP Calculus BC Unit 10: Infinite Sequences and Series

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Unit 10: Infinite Sequences and Series

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Question 1

Find the Lagrange error bound for approximating $sin (0.1)$ using the third-degree Taylor polynomial about $x = 0$ .

\frac{( 0.1 ) ^{4}}{4 !}

\frac{( 0.1 ) ^{3}}{3 !}

\frac{( 0.1 ) ^{5}}{5 !}

\frac{( 0.1 ) ^{4}}{4}

Question 2

Use the Maclaurin series $sin x = x - \frac{x ^{3}}{3 !} + \frac{x ^{5}}{5 !} - \dots$ to find $x \to 0 lim \frac{sin x - x}{x ^{3}}$ .

- \frac{1}{6}

0

\frac{1}{6}

1

Question 3

Does the $p$ -series $n = 1 \sum \infty \frac{1}{n ^{3/2}}$ converge or diverge?

A.Diverges, because terms go to

0

B.Diverges, because

p = 3/2 > 1

C.Converges, because

p = 3/2 > 1

D.Converges, because

p = 3/2 < 2

Question 4

Does the geometric series $n = 0 \sum \infty (\frac{2}{3})^{n}$ converge? If so, find its sum.

A.DivergesB.Converges to

\frac{2}{3}

C.Converges to

3

D.Converges to

2

Question 5

Use the Ratio Test to determine whether $n = 1 \sum \infty \frac{n !}{2 ^{n}}$ converges or diverges.

A.Converges by Ratio TestB.Diverges by Ratio Test (limit

= \infty > 1

)C.Inconclusive (limit

= 1

)D.Converges by comparison with

\sum 1/ 2^{n}

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Question 6

Find the interval of convergence of the power series $n = 0 \sum \infty \frac{x ^{n}}{n + 1}$ .

(- 1, 1)

[- 1, 1]

[- 1, 1)

(- \infty, \infty)

Question 7

Does the alternating series $n = 1 \sum \infty \frac{( - 1 ) ^{n + 1}}{n ^{2}}$ converge?

A.No, because terms are not all positive.B.Yes, absolutely (since

\sum 1/ n^{2}

converges).C.Conditionally convergent only, not absolutely.D.Yes, but only conditionally (alternating series test).

Question 8

Find the Maclaurin series for $f (x) = e^{- x}$ up to the $x^{3}$ term.

1 - x + \frac{x ^{2}}{2} - \frac{x ^{3}}{6} + \dots

1 + x + \frac{x ^{2}}{2} + \frac{x ^{3}}{6} + \dots

- 1 + x - \frac{x ^{2}}{2} + \frac{x ^{3}}{6} + \dots

x - \frac{x ^{2}}{2} + \frac{x ^{3}}{3} - \dots

Question 9

Find the Taylor series for $ln (1 + x)$ centered at $x = 0$ (Maclaurin series).

n = 1 \sum \infty \frac{x ^{n}}{n}

n = 1 \sum \infty \frac{( - 1 ) ^{n + 1} x ^{n}}{n}

n = 0 \sum \infty \frac{( - 1 ) ^{n} x ^{n}}{n !}

n = 1 \sum \infty \frac{( - 1 ) ^{n} x ^{n}}{n}

Question 10

The Maclaurin series for $\frac{1}{1 - x}$ is $\sum_{n = 0}^{\infty} x^{n}$ . Use this to find a series for $\frac{x}{( 1 - x ) ^{2}}$ .

n = 1 \sum \infty n x^{n}

n = 0 \sum \infty x^{n + 1}

n = 0 \sum \infty (n + 1) x^{n}

n = 0 \sum \infty n x^{n - 1}

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Question 11

The Taylor polynomial of degree $4$ for $f (x) = cos x$ about $x = 0$ is $P_{4} (x) = 1 - \frac{x ^{2}}{2} + \frac{x ^{4}}{24}$ . Use the Lagrange error bound to estimate $∣ cos (0.5) - P_{4} (0.5) ∣$ .

\frac{( 0.5 ) ^{5}}{120}

\frac{( 0.5 ) ^{4}}{24}

\frac{( 0.5 ) ^{6}}{720}

\frac{( 0.5 ) ^{5}}{5}

Question 12

Determine the radius of convergence of $n = 0 \sum \infty \frac{( x - 3 ) ^{n}}{4 ^{n} ( n + 1 ) ^{2}}$ .

R = 1

R = 4

R = 16

R = \infty

AP Calculus BC Unit 10: Infinite Sequences and Series — Free with Answer Explanations | Test Practice Hub